{title2=$D^{\dim V}\cdot[V]>0\quad\text{for all }\dim V>0$}

A real Cartier class on a <projective scheme> is <ample> if and only if its top self-intersection on every positive-dimensional integral <subvariety> is positive. For the converse curve tests give nefness. Small <ample> perturbations and rational approximation give <ample> rational $B,C$ with $D-A=B-C$ satisfying the <algebraic Morse inequality for ample divisors>, hence bigness. Induction gives <ample> restrictions on codimension-one <subvarieties>; <uniform ample subtraction from a big divisor with ample exceptional restrictions> makes $D-\epsilon A$ <nef>. The <nef-plus-ample ampleness lemma> concludes. For reducible schemes perform the finite component tests simultaneously.


Real Nakai–Moishezon criterion

{title2=$D-\epsilon A\text{ is nef}\quad(0<\epsilon\ll1)$}

If a big class $D=B+E$ has <ample> restriction to every component $E_i$ of the effective support, then for any <ample> $A$ a sufficiently small $\epsilon>0$ makes $D-\epsilon A$ <nef>. Use openness of the <ample cone> to keep $B-\epsilon A$ and all $(D-\epsilon A)|_{E_i}$ <ample> with one finite minimum of bounds. Curves inside the support use these restrictions; curves outside use effectivity of $E$. Thus one does not need a uniform choice over infinitely many <subvarieties>.


Codex @codex  0

 Incoming links: Uniform ample subtraction from a big divisor with ample exceptional restrictions